Analysis of an M/M/1 queueing system with impatient customers and a variant of multiplevacation policy

Dequan Yue; Wuyi Yue; Zsolt Saffer; Xiaohong Chen

doi:10.3934/jimo.2014.10.89

Article Contents

2014, Volume 10, Issue 1: 89-112. Doi: 10.3934/jimo.2014.10.89

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Analysis of an M/M/1 queueing system with impatient customers and a variant of multiple vacation policy

1.
Department of Statistics, College of Sciences, Yanshan University, Qinhuangdao 066004
2.
Department of Intelligence and Informatics, Konan University, 8-9-1 Okamoto, Kobe 658-8501
3.
Department of Telecommunications, Budapest University of Technology and Economics, Budapest
4.
College of Science, Yanshan University, Qinhuangdao 066004

Received: August 31, 2012

Revised: May 31, 2013

Published: December 2013

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Abstract

In this paper, we consider an M/M/1 queueing system with impatient customers and a variant of multiple vacation policy, where we examine the case that customer impatience is due to the servers' vacation. Whenever a system becomes empty, the server takes a vacation. However, the server is allowed to take a maximum number $K$ of vacations if the system remains empty after the end of a vacation. This vacation policy includes both a single vacation and multiple vacations as special cases. We derive the probability generating functions of the steady-state probabilities and obtain the closed-form expressions of the system sizes when the server is in different states. We further make comparisons between the mean system sizes under the variant vacation policy and the mean system sizes under the single vacation policy or the multiple vacation policy. In addition, we obtain the closed-form expressions for other important performance measures and discuss their monotonicity with respect $K$. Finally, we present some numerical results to show the effects of some parameters on some performance measures.

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Mathematics Subject Classification: Primary: 60K25; Secondary: 90B22.

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